Lecture 5 Circuits Topics KVL KCL Op Amps Thevenin equivalents Lab Exercises build robot head Motor servo controller rotating neck Phototransistor robot eyes Integrate to make a light tracking system The Circuit Abstraction Circuits represent systems as connection of elements Through while currents through variables ow Across while voltages across variables develop We can represent the ashlight as a voltage source a battery connected to a resistor a light bulb The voltage source generates a voltage v across the resistor and a current i through the resistor We can represent the ow of water by a circuit Flow of water into and out of tank are represented as thought variables r and r respectively Hydraulic pressure at bottom of tank is represented by the across variable P gh Circuits are important for two very different reasons As a physical system Power from generators and transformers to power lines Electronics from cell phones to computers As models of complex systems Neurons Brain Cardiovascular system Hearing Circuits are the basis of the enormously successful semiconductor industry Circuits as models of complex systems myelinated neutron The primitives are the elements Sources capacitors resistors The rules of combination are the rules that govern Flow of current through variable Development of voltage Analyzing Circuits We will start with the simplest elements resistors and sources Analyzing simple circuits is straightforward Example 1 The voltage source determines the voltage across the resistor v 1V so the current through the resistor is i v R 1A Example 2 The current course determines the current through the resistor i 1A so the voltage across the resistor is v iR 1V Check yourself Analyzing simple circuits Correct answer 1 The current through the resistor is 1A Analyzing Circuits KVL The sum of the voltages around any closed path is zero Example v v v 0 Check yourself Kirchhoff s Voltage Law Correct answer 5 How many KVL relations are there for this circuit There are 7 loops Planar circuits can be characterized by their inner loops KVL equations for the inner loops are independent A v v v 0 B v v v 0 C v v v 0 A B 0 All possible KVL equations for planar circuits can be generated by combinations of the inner loops One KVL equation can be written for every closed path in a circuit Sets of KVL equations are not necessarily linearly independent More complex circuits can be analyzed by systematically applying Kirchhoff s voltage law KVL and Kirchhoff s current law KCL KVL equations for the inner loops of planar circuits are linearly independent The ow of electrical current is analogous to the ow of incompressible uid such as water Current i ows into a node and two currents i and i ow out The net ow of electrical current into or out of a node is zero The net current out of the top node must be zero Analyzing Circuits KCL i i i i i i 0 Electrical currents cannot accumulate in elements so current that ows into a circuit element also ows out Check yourself Kirchhoff s Current Law Correct answer 2 There are three equations that can be written but only two of them are linearly independent KCL equations Check yourself Kirchhoff s Current Law Correct answer 1 There are three linearly independent KCL equations General form the number of linearly independent KCL equation is the number of nodes minus one The net current out of any closed surface which can contain multiple nodes is zero KVL KCL and Constitutive Equations Circuits can be analyzed by combining All linearly independent KVL equations All linearly independent KCL equations One constitutive equation for each element The node method is one of many ways to systematically reduce the number of circuit equations and unknowns Label all nodes except one ground 0 V Write KCL for each node whose voltage is not unknown Solve for the unknowns The loop current method is another way to systematically reduce the number of circuit equations and unknowns Node Voltages Loop Currents Label all the loop currents Write KVL for each loop Solve for the unknowns Check yourself Current Correct answer 3 The current in the circuit is 1A Common Patterns Circuits can be simpli ed when two or more elements behave as a single element A one port is a circuit that can be represented as a single element A one port has two terminals Current enters one terminal and exits another producing a voltage v across the terminals Series Combinations Parallel Combinations The series combination of two resistors is equivalent to a single resistor whose resistance is the sum of the two original resistances R R R The resistance of a series of combination is always larger than either of the original resistances The parallel combination of two resistors is equivalent to a single resistor whose conductance 1 resistance is the sum of the two original conductances The resistance of a parallel combination is always smaller than either of the original resistances Check yourself Equivalent Resistance Correct answer 3 The equivalent resistance of the one port is 2 ohms Voltage Divider Current Divider Resistors in series act as voltage dividers Resistors in parallel act as current dividers Interaction of Circuit Elements Circuit design is complicated by interactions among the elements Adding an element change voltages and currents throughout the circuit Example closing a switch is equivalent to adding a new element Check yourself Closing a switch Correct answer 2 Closing the switch cause V to decrease and I to increase One Ports If a circuit connects to the world via just two terminals then that circuit can be represented by a single generalized element called a one port regardless of how many components are in the circuit This is analogous to replacing delays gains and adders with a system function combining a sequence of operations in a procedure call combining diverse data in a list These representations are compositional they replace multiple elements with a single element that can be used in the same way that primitives are used Current Voltage Relations Current voltage relations for resistors and sources are straight lines Current voltage relations are analogous to system functionals They summarize the behavior of a one port Parallel One Ports Series One Ports If the i v curves for two one ports are both straight lines then the i v curve for the parallel combination is a straight line This is because the sum of two straight lines is a straight line If the i v curves for two
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